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I have tried to add a text widget to the right column with an "inline list" to display my social icons but it doesn't work. Each list item is given it's own line.

I also have a social media plugin (Share and Follow) that adds icon to the end of each post for readers to share. They are to be displayed as "inline" list items as well but again are given their own line.

List items will always appear on a new line unless you use CSS to make them display inline or float to the left. The issue is probably conflicting CSS. Use Firebug to see what CSS is being used on the elements in question.